On compactness of minimizing sequences subject to a linear di erential constraint

For Ω ⊂ RN open (and possibly unbounded), we consider integral functionals of the form F (u) := ∫ Ω f(x, u) dx, de ned on the subspace of Lp consisting of those vector elds u which satisfy Au = 0 on Ω in the sense of distributions. Here, A may be any linear di erential operator of rst order with constant coe cients satisfying Murat's condition of constant rank. The main results provide sharp conditions for the compactness of minimizing sequences with respect to the strong topology in Lp.